2.03 add math module 03 quadratic equations · pdf file1.12.2008 · spm quadratic...

QUADRATIC

ADDITIONAL MMOD

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ATHEMATICS

EQUATIONS

ULE 3

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1

CHAPTER 2 : QUADRATIC EQUATIONS

MODUL 3

2.1 CONCEPT MAP 2

2.2 GENERAL FORM 3

2.2.1 Identifying 3Exercises 1 3

2.2.2 Recognising general form of quadratic equation 4.ax2 + bx – c = 0Exercises 2 4

2.3 SOLVING QUADRATIC EQUATIONS 6

2.3.1 Factorisation 6Exercises 1 6

2.3.2 Completing the square 8Exercises 1 8

2.3.3 Quadratic formula 10Exercises 1 10

2.4 PASS YEARS QUESTIONS 12

2.5 ASSESSMENT 13

ANSWERS 15

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MODUL 3

2.1 CONCEPT MAP

QUADRATIC EQUATIONS

x x = 0

xx 2 = 0

> = 0 (Positive)

Two differentroots

http://m


GENERAL FORM

2

ROOTSx = ,

.ax2 + bx + c = 0

b2 – 4ac

roots

< = 0 (n

Noro

Types of roots

athsmozac.blogspot.co

Factorization

Completing the square

Formula

egative)

realots

= 0

TwoEqual

m

3

2.2 GENERAL FORM

2.2.1 Identifying

Example 1

4x + 3 = 2x2x + 3 = 0

The highest power of variable x is 1Therefore 4x + 3 = 2x is not a quadraticequation

Example 2

x(x + 5) = 7x2 + 5x - 7 = 0

The highest power of variable x is 2Therefore x(x + 5) = 7 is a quadraticequation

Exercises 1Identify which of the following are quadratic equation

1. 3 =x2

5.2. x(2x + 3) = x - 7

3. ( x + 4 )(2x – 6) + 3 = 0 4. (3m + 5)2 = 8m

5. x (7 - 2x + 3x2) = 0 6. 3x2 – 5 = 2x( x + 4)



4

2.2.2 Recognising general form of quadratic equation .ax2 + bx – c = 0

Example 1

.x2 = 5x – 9

.x2 – 5x + 9 = 0

Compare with the general form.ax2 + bx – c = 0

Thus, a = 1, b = -5 and c = 9

Example 1

4x =x

xx 22

4x(x) = x2 – 2x4x2 - x2 – 2x = 03x2 – 2x = 0

Compare with the general formThus, a = 3, b = - 2 and c = 0

Exercises 2Express the following equation in general form and state the values of a, b and c

1. 3x =x2

5

.2. (2x + 5) =x

7

3. x( x + 4 ) = 3 .4. (x – 1)(x + 2) = 3



5

5.x

4=

x

x

5

3 6. x2 + px = 2x - 6

7. px (2 – x) = x – 4m 8. (2x – 1)(x + 4) = k(x – 1) + 3

9. (7 – 2x + 3x2) =3

1x10. 7x – 1 =

x

xx 22



6

2.3. SOLVING QUADRATIC EQUATIONS

2.3.1 Factorisation

Exercises 3Solve the following quadratic equation by factorisation

1. x2 + 3x - 4 = 0 2. x2 -2x = 15

3. 4x2 + 4x – 3 = 0 4. 3x2 - 7x + 2 = 0

Example 1

.x2 + 6x + 5 = 0 x + 3 3x( x + 3)(x + 2) = 0 .x + 2 2x.x + 3 = 0 or x + 2 = 0

.x = -3 x = - 2 .x2 + 6 5x

Therefore, The roots of the equation are.x = -3 and -2

Example 2

4(x +3) = x(2x – 1)4x + 12 = 2x2 - x2x2 - 5x - 12 = 0(2x + 3)(x - 4) = 02x + 3 = 0 or x - 4 = 0

.x =2

3x = 4

Therefore, The roots of the equationare

.x =2

3and 4



7

5. x2 = 3x – 2 6. x(2x - 5) = 12

7. 8x2 + x = 21(1 – x) 8. (2y – 1)(y + 4) = -7

9. 4y -y

1= 3 10.

23

67

m

m= m



8

2.3.2. Completing the square

Example 1

.x2 – 6x + 7 = 0.x2 - 6x = -7

.x2 – 6x +2

2

6

= - 7 +

2

2

6

.x2 – 6x + (-3)2 = -7 + (-3)2

(x - 3)2 = 2

.x – 3 = 2

.x = 3 2

.x = 3 + 2 or 3 - 2

.x = 4.414 or 1.586

Example 2

2x2 -5x – 1 = 02x2 – 5x = 1

.x2 -2

5x =

2

1

.x2 -2

5x +

2

4

5

=

2

1+

2

4

5

x =

2

1+

16

25

=16

33

.x -4

5=

16

33 =

4

33

.x =4

5+

4

33or

4

5-

4

3

.x =4

335 or

4

335

.x = 2.686 or -0.186

Exercises 4Solve the following quadratic equation by completing the square

1. (x + 3 )2 = 16 2. (5x - 4)2 = 24

Rearrange in theform.x2 + px = q

Add

2

2

...

xoftcoefficien

To both sides



Change thecoefficient

2

2

4

5

3

of x to 1

9

3. x2 - 8x + 12 = 0 4. 3x2 + 6x – 2 = 0

5. 5x2 – 7x + 1 = 0 6. 2x2 – 3x – 4 = 0

7. (x + 1)(x - 5) = 48. 1 -

x

1=

22

3

x



10

Quadratic formula

Example 1

.x2 + 5x + 2 = 0.a = 1, b = 5, c = 2

Using the formula x =a

acbb

2

42

.x =

12

21455 2

.x =2

8255 =

2

175

.x =2

175 or

2

175

= - 0.438 or - 4.562

Example 2

3x2 = 4x + 23x2 - 4x – 2 = 0.a = 3, b = -4 , c = - 2

Using the formula

.x =

32

23444

=

6

24164

=6

404

.x =6

404 or

6

404

= 1.721 and – 0.387

Exercises 5Solve the following quadratic equations by using the quadratic formula

1. x2 – 11x + 28 = 0 2. –x2 – 3x + 5 = 0



11

3. 2x2 + 11x + 9 = 0 4. 3x2 + 14x – 9 = 0

5. 10x(2x – 1) – 8 = x(2x + 35) 6. (x – 1)(4x – 9) + 7 = 10x

7.3

211

v

v= 2v 8.

1

132

2

xx

xx= 2



12

9.5

23 x+

2

3

x

x= 1 10.

x

x1+ 3 =

5

7

x

2.4 PAST YEARS QUESTIONS

SPM 2001. PAPER 1 Question 3

1. Solve the quadratic equation 2x(x + 3) = ( x + 4)(1 - x). Give your answercorrect to four singnificant figures.

SPM 2003. PAPER 1 Question 3

1. Solve the quadratic equasion 2x(x – 4) = (1 – x)(x + 2)Give your answer correct to four significant figures.



13

2.5 ASSESSMENT ( 30 minutes)

1. Express 3x2 – 2px = 5x - 7p in genegal from

2. Find the roots of the equation 2x2 + 5x = 12

3. Find the roots of2

1

x=

3

x,

4. By using the quadratic formula, solve the equation 2x2 – 5x – 1 = x(4x - 2)



14

5. Solve the quadratic equation (5x – 3)(x + 1) = x(2x – 5) .Give your answer correct to four significant figures.

6. Given the equation x2 + 4x – 5 = (x – a)2 + b , find the values of a and b



15

ANSWERS

Exercises 11. No 2. Yes3. Yes 4. Yes5. No 6. Yes

Exercises 21. 6x2 – 5 = 0 a= 6, b = 0, c = -52. 2x2 + 5x – 7 = 0 a = 2 , b = 5 , c = -73. x2 + 4x – 3 = 0 a = 1, b = 4 , c = - 34. x2 + x – 5 = 0 a = 1, b = 1, c = -55. x2 + 7x – 20 = 0 a = 1, b = 7, c = -206. x2 + (p – 2)x + 6 = 0 a= 1, b=(p – 2), c= 67. px2 + (1 – 2p)x – 4m = 0 a = p , b = (1 – 2p) , c = -4m8. 2x2 + (7 - k) x + k – 7 = 0 a = 2, b = (7 - k), c = (k – 7)9. 9x2 - 7x + 20 = 0 a = 9, b = -7 c = 2010. 6x2 + x = 0 a = 6, b = 1, c = 0

Exercises 31. 1, -4 2. 5, -3

3.2

1, -

2

34.

3

1, 2

5. 1, 2 6. 4,2

3

7.2

7,

4

37.

2

1, - 3

9.4

1, 1 10. 1,

3

7

Exercises 41. 1, -7 2. 1.77, - 0.173. 2, 6 4. 0.457 , - 1.4375. 0.161, 1.239 6. 2.351 , - 0.8517. 5.60, -1.60 8. 1.823, -0.823



16

Exercises 51. 4, 7 2. 1.191, - 4.191

3. -1,2

94. 0.573 , -5.239

5. 2.667, - 0.169 6. 0.810, 4.94

7. 2,2

18. 5.192, -1.925

9. 2.812, - 0.119 10. -0.403, -3.069

PAST YEARS QUESTIONS

1. 0.393, -3.393 2. 2.591, -0.2573

ASSESSMENT

1. 3x2 – (2p + 5)x + 7p = 0

2.2

3, 4

3. – 3, 14. - 0.5, -15. 0.370, -2.706. a = 2, b = -9



17

SPM

QUADRATIC EQUATIONS

ADDITIONAL MATHEMATICSMODULE 4



18


MODUL 4

2.1 CONCEPT MAP 2

2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS and 3

2.2.1 Form a quadratic equation with the roots 4

Exercises 1 4

2.2.2 Determine the sum of the roots and product of the roots of the 5following quadratic equations.Exercises 2 5

2.3. TYPES OF ROOTS QUADRATIC EQUATION 6

2.3.1 Determine the types of roots for each of the following 6quadratic equationsExercises 3

2.4 SOLVING PROBLEMS INVOLVING (.b2 - 4ac ) 7

2.4.1 Find the values of k for each of the following quadratic 7equations which has two equal roots

Exercises 4 7

2.4.2 Find the range of values of h for each of the following 8quadratic equations which roots are different

Exercises 5 8

2.4.3 Find the range of values of m for each of the following 9quadratic equations which has no roots

Exercises 6 9

2.5 PASS YEARS QUESTIONS 10

2.6 ASSESSMENT (30 MINUTES) 12

ANSWERS 14




MODUL 4

2.1 CONCEPT MAP

S

QUADRATIC EQUATIONS

.x =

GENERAL FORM

Sum

Pro

TY

http://mathsm


ROOTS
and x =
The quadratic equation

FORMING A QUADRATIC EQUATIONFROM GIVEN ROOTS

of the roots

duct of the roots

The quadratic equation

19

PES OF ROOTS

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20

2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS and

Notes

If x1 = a and x2 = bThen (x – a) = 0 and (x – b) = 0

(x – a)(x – b) = 0.x2 – bx – ax + ab = 0.x2 – (a + b)x + ab = 0

The quadratic equation with roots and is written as

.x2 – ( )x + = 0 …….(1)

From general form ax2 + bx + c = 0

a

ax 2

+a

bx+

a

c= 0

.x2 +a

bx+

a

c= 0 ……………

Compare with the equations (1) and (2) .x2 – (

x2 +a

bx+

– ( ) =c

b

( ) = -c

b( The sum of the root

=a

c( The product of the

Sumof theroots

Product.of theroots

Product.of the

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Sumof the

………(2)

)x + = 0 ……….….(1)

a

c= 0 ……………………(2)

s)

roots)

roots roots

ot.com

21

2.2.1 Form a quadratic equation with the roots

Exercises 1Roots and

Sum of.the roots( )

Product of.the roots

The puadratic equation

.x2 – ( )x + = 0

Example 13 , 2 5 6 x2 – (5)x + 6 = 0

Example 2

4

1, - 3

4

1+ (-3)

=4

121=

4

11

4

1(- 3) =

4

3.x2 –

4

11x +

4

3= 0

4x2 + 11x – 3 = 0

a) 4 , -7

b)

2 ,3

1

c)

3

1,

2

1

d)

5

1,

3

2

e)

3k,5

6k



22

2.2.2 Determine the sum of the roots and product of the roots of the followingquadratic equations.

Exercises 2The puadratic equation Sum of

.the rootsProduct of.the roots

Example 1.x2 – 6x + 9 = 0 6 9

Example 1.9x2 + 36x - 27 = 0

9

9 2x+

9

36x-

9

27= 0

.x2 + 4x – 3 = 0

-(4) = -4 -3

a) .x2 + 73x - 61 = 0

b) 7x2 - 14x - 35 = 0

c) 2x(x + 3) = 4x + 7

d) 2x +x

2=

4

1

e)4x2 + kx + k – 1 = 0



2.3. TYPES OF ROOTS QUADRATIC EQUATION

2.3.2 Determine the ty

Exercises 31. 2x2 - 8x + 3 = 0

3. 3x2 = 7x - 5

Example 1

a) . x2 – 12x + 27 = 0. a = 1, b = -12 , c = 27

.b2 - 4ac = (-12)2 – 4(1)(27= 144 – 108= 36 > 0

Thus, x2 – 12x + 27 = 0Has two different roots

. b2 - 4ac > 0

. Two different

ht


pes o

)

E

b.

.b

.x =

a

acbb

2

42

.b2 - 4ac

. b2 - 4ac < 0

tp://

. b2 - 4ac = 0

23

f roots for each of the following quadratic equations

2. -2y2 + 6x + 3 = 0

4. 4x2 = a(4x - a)

xample 2

) .4 x2 – 12x + 9 = 0a = 4, b = -12 , c = 9

2 - 4ac = (-12)2 – 4(4)(9)= 144 - 144= 0

Thus, 4x2 – 12x + 9 = 0Has two equal roots

Example 3

c) .2 x2 – 7x + 10 = 0. a = 2, b = -7 , c = 10

.b2 - 4ac = (-7)2 – 4(2)(10)= 49 – 80= - 31< 0

Thus, 2x2 – 7x + 10 = 0Has no real roots

. Two equal root . No real roots

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24

2.4 SOLVING PROBLEMS INVOLVING (.b2 - 4ac )

The discriminant b2 - 4ac of the quadratic equation can be used toa) find an unknown value in an equation

Example 1

The quadratic equation x2 - 2px + 25 = 0Has two equal roots. Find the value p

x2 - 2px + 25 = 0Thus, a = 1, b = -2p , c = 25Using .b2 - 4ac = 0

(-2p)2 – 4(1)(25) = 04p2 - 100 = 0

4p2 = 100p2 = 25

p = 25

p = 5

Example 2

The quadratic equation x2 – 2kx = -(k – 1)2

Has no roots. Find the range of values of k

x2 – 2kx = -(k – 2)2

x2 – 2kx + (k – 2)2 = 0Compare with ax2 + bx + c = 0Thus .a = 1, b = - 2k , c = (k – 2)2

Using b2 - 4ac < 0(-2k)2 – 4(k2 -2k + 1)< 04k2 - 4k2 + 8k – 4 < 0

8k – 4 < 08k < 4.k < 4

2.4.1 Find the values of m for each of the following quadratic equations whichhas two equal roots

Exercises 4

1. mx2 - 4x + 1 = 0 2. x2 – 6x + m = 0

3. x2 – 2mx + 2m + 3 = 0 4. x2 - 2mx - 4x + 1 = 0



25

5. x2 + 6x - 9 = m (2x - 3) 6. x2 + 2(x + 2) = m(x2 + 4)

2.4.2 Find the range of values of h for each of the following quadratic equationswhich roots are different

Exercises 5

1. x2 - 6x - h = 0 2. hx2 – 4x – 3 = 0

3. x2 + 6x + h + 3 = 0 8. 2hx2 + 4x + 1 = 0



26

5. x(5 – 2x) = h + 2 6. 2(hx2 – 1) = x(x – 6)

2.4.3 Find the range of values of m for each of the following quadratic equationswhich has no roots

Exercises 6

1. 2x2 + 2x - m = 0 2. mx2 + 3x - 3 = 0

3. x2 + 2x + m - 3 = 0 4. 3x2 + 1 = 2(m + 3x)



27

5. (h + 1)x2 + 8x + 6 = 0 6. (m - 3)x2 + 2(1 – m)x = -(m + 1)

2.5 PASS YEARS QUESTIONS

SPM 2001/P1 Question 4

1. Given that -1 and h are roots of the quadratic equation(3x – 1)(x – 2) = p(x – 1), where p is a constant, find the values of h and p


2. Given that 3 and n are roots the equation (2x + 1)(x – 4) = a(x – 2),where a and n are constants, find the values of a and n.



28


3. It is given that3

and

3

are roots of the quadratic equation px(x – 1) = 3q + x

If = 12 and = 3, find the values of p and q


4. Form a quadratic equation which has the roots – 5 and4

3,

Give your answer in the form of ax2 + bx + c = 0 ,Where a, b, and c are constans



29

2.6 ASSESSMENT (30 MINUTES)

1. Form a quadratic equation which has the roots 3 and - 4

2. Given that -3 and 4 are roots of the quadratic equation x2 + ax = bFind the values of a and b

3. The quadratic equation x2 - kx + 2k = 4 has roots 2 and 6Find the values of k



30

.4 Find the values of h if the equation x2 = 4hx - 36 has equal roots

5. The quadratic equation kx2 - 2(3 + k)x = 1 – k has no real roots.Find the range of vales of k

6. The quadratic equasion x(x – 2m) = - ( 3m + 4) has equal roots, finda) the value of mb) the roots



31

ANSWERS

Exercises 1

a. -3, -28, x2 + 3x – 28 = 0 b.3

5,

3

2, , 3x2 -5x – 2 = 0

c.6

5,

6

1, 6x2 - 5x + 1 = 0 d.

15

7,

15

2, 15x2 + 7x – 2 = 0

e.5

21k,

5

18 2k, 5x2 - 21kx – 18 k2 = 0

Exercises 2

a. -73 , - 61 b. 2, - 5

c. -1,2

7, d.

8

1, 1

e.4

k,

4

1k

Exercises 3

1. Two different roots 2. Two different roots3. No real roots 4. Two equal roots

Exercises 4

1. m = 4 2. m = 93. m = 3 and -1 4. m = - 3 and -1

5. m = 3 and 6 6. m =2

3and

2

1

Exercises 5

1. h > - 9 2. h >3

4

3. h < 6 4. h < 2

5. h <8

96. h >

4

7



32

Exercises 5

1. m <2

12. m <

4

3

3. m > 4 4. m < -3

5. m >3

56. m >

3

1

PASS YEARS QUESTIONS

1. P = - 6 , h =3

4

2. a = -7 n = 3

3. p =3

1, q =

27

1

4. 4x2 + 17x – 15 = 0

ASSESSMENT

1. x2 + x - 12 = 02. a = -1, b = 123. k = 84. h = 3

5. k <7

9

6. m = 4, -1x = 4, -1



2.03 add math module 03 quadratic equations · pdf file1.12.2008 · spm quadratic...

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